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Russian Mathematics

, Volume 63, Issue 1, pp 14–23 | Cite as

Approximation of Non-Analytic Functions by Analytical Ones

  • H. H. BurchaevEmail author
  • G. Yu. RyabykhEmail author
Article

Abstract

We study the properties of the elements of best approximation for functions defined in the unit disk by functions from the Bergman space. For functions of a special type, we find a sufficiently accurate description of the properties of these elements in terms of the Hardy and Lipschitz classes. The obtained result is based on an analysis of the corresponding duality relation for extremal problems. The developed method is also applicable to relatively smooth (in terms of Sobolev spaces) approximated functions.

Key words

Bergman space Hardy space best approximation linear functional extremal problem 

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Notes

Acknowledgments

Supported by Russian Foundation for Basic Research, grant No. 18-01-00017.

References

  1. 1.
    Khavinson, D., Shapiro, H. S. “Best Approximation in the Supremum Norm by Analytic and Harmonic Functions”, Arc. Mat. 39, No. 2, 339–359 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burchayev, H. H., Ryabykh, V. G., Ryabykh, G. Yu. Integro-Differential Operators in Spaces of Analytic Functions With Aapplications (ITs DGTU, Rostovon-Don, 2014) [in Russian].Google Scholar
  3. 3.
    Pozharskii, D. A., Ryabykh, V. G., Ryabykh, G. Yu. Integral Operators in Spaces of Analytic and Similar to Them Functions (ITs DGTU, Rostovon-Don, 2011) [in Russian].Google Scholar
  4. 4.
    Garnett John B. Bounded Analytic Functions (Springer, 1981).zbMATHGoogle Scholar
  5. 5.
    Ferguson, T. “Extremal Problems in Bergman Spaces and Extension of Ryabykh’s Theorem”: A dissertation submitted by partial fulfillment of the requirement for the degree of Doctor of Philosophy (Math.) (Univ. of Michigan, Michigan, 1–76, 2011).Google Scholar
  6. 6.
    Hoffman, K. Banach Spaces of Analytic Functions (Dover Publications, N. Y., 1962).zbMATHGoogle Scholar
  7. 7.
    Kantorovitch, L. V., Akilov, G. P. Functional Analysis (Nauka, Moscow, 1984) [in Russian].Google Scholar
  8. 8.
    Goluzin, G. M. Geometric Theory of Functions of Complex Variable (Gostekhizdat, Moscow-Leningrad, 1952) [in Russian].zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Chechen State UniversityGroznyRussia
  2. 2.Don State Technical UniversityRostov-on-DonRussia

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